Answer :
The Large Counts condition is met (np >= 10, n(1-p) >= 10), allowing the z-test. Other conditions are satisfied, validating the test.
Therefore, the correct answer is: **d. The test cannot be performed because the Large Counts condition has not been met.**
To perform a one-sample z-test for the population proportion, we need to check certain conditions:
1. **Random Sample**: The sample should be chosen randomly from the population to ensure that it is representative.
2. **Normality of Population**: There are two criteria for this condition:
- If the population distribution is approximately normal, we can proceed.
- If the population distribution is not normal, the sample size should be large enough (n x p >= 10 and n x (1-p) >= 10) to ensure that the sampling distribution of the sample proportion is approximately normal by the Central Limit Theorem.
3. **Independence**: Each observation in the sample should be independent of the others.
Given:
- Sample size (n) = 30
- Population proportion (p) = 0.04
We need to check if these conditions are met.
a. **We can’t determine if the conditions have been met until we have the sample proportion, p hat.**
- This statement is incorrect because we can still check the conditions without knowing the sample proportion.
b. **The test cannot be performed because the Random condition has not been met.**
- This statement is incorrect because the problem states that the buyer took a random sample of 30 apples, so the Random condition has been met.
c. **All conditions for performing the test have been met.**
- This statement is incorrect because we haven't checked the Large Counts condition yet.
d. **The test cannot be performed because the Large Counts condition has not been met.**
- To check the Large Counts condition, we need to verify whether the sample size is large enough to ensure that the sampling distribution of the sample proportion is approximately normal.
- The Large Counts condition is met if both np >= 10 and n(1-p) >= 10.
- Here, n = 30 and p = 0.04
- np = 30 x 0.04 = 1.2
- n(1-p) = 30 x (1 - 0.04) = 28.8
- Both np and n(1-p) are greater than or equal to 10.
- So, the Large Counts condition is met.
Therefore, the correct answer is: **d. The test cannot be performed because the Large Counts condition has not been met.**
Final answer:
The correct statement about the conditions for performing a one-sample z test for the population is that we can't determine if the conditions have been met until we have the sample proportion, p hat.
Explanation:
The correct statement about the conditions for performing a one-sample z test for the population is:
a. We can’t determine if the conditions have been met until we have the sample proportion, p hat.
When performing a hypothesis test of a single population proportion, the conditions for a binomial distribution need to be met. These conditions include a certain number of independent trials, success or failure outcomes, and each trial having the same probability of success. In addition, the shape of the binomial distribution should be similar to the shape of a normal distribution, which is ensured when np and nq are both greater than five. Therefore, the conditions cannot be definitively determined until the sample proportion, p hat, is known.
